It's basic supply & demand that rent control causes housing shortages. If rent control was replaced with adequate rent subsidies for landlords (for the lower/mid class tenants) and some of the more draconian regs were eased, this problem would be solved overnight. Politically however, this is probably impossible and/or extremely controversial, but I believe this arrangement would be better for all tenants.
Yup. Simply disassociating the benefits of rent control from the housing unit it is attached to would go a long way to permitting development. The way things are today, rent control basically serves as an almost permanent mutex on the development of the underlying real estate.
I'd be more useful if cities made it a policy that the amount of housing subsidy you receive is proportional to the year you moved to the city. That would achieve the same goal as rent control (i.e. affordable housing), but alleviates the mutex on many structures that could be torn down to make way for structures that permit great density.
Affordable housing programs are fine. Tying up the underlying asset indefinitely is counterproductive.